Maya Number System and the Concept of Zero

The Maya numeral system was the only fully positional number system ever developed in the pre-Columbian Americas, and one of only a handful of independent inventions of true place-value arithmetic in human history. Built on a base of 20, written with just three signs (a dot for one, a bar for five, and a shell for zero), it could in principle record any integer, however large. The question of where, when, and how the Maya arrived at it, and how the concept of zero fit into it, is treated on the related pages Did the Maya really invent the concept of zero?. This page covers the system itself: its signs, its structure, its use in the calendar, and the multiplication tables that made it work.

The three signs: dot, bar, and shell

Three graphic signs carried the entire range of Maya numbers from zero to nineteen. A single dot represented one; a horizontal bar represented five; a stylized cowrie or nautilus shell — sometimes read as a half-flower or an open maw — represented zero. Numbers from one to four were written as one to four dots; five to nine as one or more bars topped with dots (e.g. 7 = one bar of 5 plus two dots); ten to nineteen as two bars plus the appropriate number of dots. The full set of digit signs is reproduced in nearly every introduction to Maya writing, language, and books and in the encyclopedia entries on Maya hieroglyphs and the glyph system.

Larger numbers were built by stacking these digit signs vertically. Each row was a higher power of the base. The bottom row recorded units (the multiplier 1), the second row recorded 20s, the third 360s (or 400s, depending on context — see below), the fourth 7,200s, and so on. The position of a digit within a column determined its value: in a place-value system, the “3” in the second row of a number meant 3 × 20, but a “3” in the third row meant 3 × 360. A dot in the third row thus had a different value from a dot in the bottom row. The shell-glyph zero allowed scribes to leave a column empty without losing the position: writing 0.5.0 in the second and fourth rows of a date-stamp meant that those columns mattered and that the system should “carry” upward correctly.

This is the practical reason the zero is so important. Without a sign for “nothing here, but keep the place,” a positional system collapses into ambiguity: the number “1.0.0” would be indistinguishable from “100”. The shell-glyph zero in the Maya script is a true placeholder and an operational concept of zero, distinct from mere “nothing.”

Head-variant numerals and the alternative number series

In addition to the dot-and-bar system, the Classic Maya used a second, ceremonial series of numeral glyphs in which each integer from zero to thirteen was represented by a stylized human or deity head. The head-variant numerals — sometimes called “anthropomorphic” numerals — were used primarily in the Supplementary Series of Initial Series inscriptions and in the Lunar Series that records the age, moon-number, and other details of the Moon on a given date. Zero in this series was a head with a fleshless jaw, often interpreted as a death’s-head or as the “empty” Earth Lord. Numbers from one through thirteen each had a distinct head, almost always with a numeric coefficient attached.

The head-variant series was thus a parallel sign system, not a competing mathematics. It carried the same numbers as the dot-and-bar series and was used in the same kind of positional count. Its importance for the historian of science is that it confirms the Maya distinguished the sign for a number from the concept of the number — a step in the direction of abstraction that the head-variant series makes visually obvious.

Vigesimal, base-20, with a calendar correction

The system is described as vigesimal because its base is 20. In a pure base-20 system every column would represent 20 times the column below it: 1, 20, 400, 8,000, 160,000, and so on. The Maya, however, used a modified version in which the third column (the 400s column of a pure vigesimal count) was instead a 360s column — that is, 18 × 20 rather than 20 × 20. From the fourth column upward the system reverted to a pure 20 multiplication. The reason is calendrical: the standard unit of the Maya Long Count, the tun, was defined as 360 days (18 × 20) to keep the day count aligned with the 365-day haab’ solar year, and a 360-day tun was the basic building block of historical time. The mixed system is summarized in the overview on The Maya Calendar System and treated in more detail on The Long Count and Calendar Round.

A typical Initial Series inscription in the modified system thus read: baktun (144,000 days, 20 × 7,200), katun (7,200 days, 20 × 360), tun (360 days, 18 × 20), uinal (20 days, 20 × 1), kin (1 day). Reading bottom-up: 1, 20, 360, 7,200, 144,000. Because the third column is 360 rather than 400, the baktun is 20 × 7,200, not 20 × 8,000 as a pure base-20 system would demand. The pure-vigesimal form survived in some Late Classic inscriptions, especially at Copan and Quirigua.

Large numbers: the Quirigua Stela E inscription

The largest numbers ever recorded by the Maya appear on the monuments of Quirigua, on the lower Motagua River in eastern Guatemala, and on the late Classic altars and stelae of the central lowlands. The record-holder is Quirigua Stela E, whose Initial Series and Supplementary Series reach into the millions of days. Its Period Number, the value multiplied out to describe a cycle beyond the 13-baktun “great cycle,” uses 33 digits in the vigesimal system. A typical line of the same monument shows a value near 1.5 billion days when projected forward, and a Period Number of 1,254,400 × 1,872,000 days — quantities that operate as easily as “12” in the Maya number scheme. The same system is recorded in the multiplication tables and on the Maya stelae, altars, and stone monuments at every major Classic site.

The Maya did not have a place-value system that ran forever in practical use; their arithmetic was for dates, calendrics, and the small integer counts of tribute, captives, and offerings. But the architecture of the system allowed it to grow in principle without limit, and a handful of scribes — most likely the dynasty’s senior aj tz’ib’ — used it freely to express durations of millions of days.

The 819-day cycle

Among the most striking uses of large numbers in Maya astronomy is the 819-day count, a period that recurs repeatedly in the inscriptions of Palenque and in the Cross Group of temples there. 819 days is, in vigesimal terms, 1.0.9 in the modified system (1 × 360 + 0 × 20 + 9 = 369, in the tun-uinal-kin column; the 819 is 369 × 360 / 162 = 819, etc., a topic discussed in detail by epigrapher Linda Schele and others). It has the remarkable property of being a common multiple of the synodic periods of the four planets the Maya tracked: 819 = 7 × 117 (where 117 is the synodic period of Mercury, of which the Maya were less certain), 819 = 3 × 273 (Saturn), 819 = 2 × 410 (Jupiter), and 819 = 1 × 819 (a Venus-Mars-Jupiter relationship). Recent reconstructions treat the 819-day cycle as a ritual-period-table that aligns the four visible planets with the 260-day tzolk’in on a long base, used in part to set the dates of the Period Ending ceremonies of Palenque’s rulers. The 819-day cycle is treated in more detail in the page on Maya astronomy.

Multiplication tables: the Dresden Codex

The single richest surviving source of Maya arithmetic is the Dresden Codex, the screenfold book held in the Saxon State Library in Dresden. Its pages 9 to 16, the so-called Venus tables, are organized in rows and columns that can be read as multiplication tables for the 584-day synodic period. The same codex contains lunar tables that work as multiplication tables for the 6-month, 11,960-day eclipse cycle. Multiplication tables proper, of the form “1 to 13, each multiplied by 7, 11, 13,” appear on pages 49 to 56, sometimes called the “multiplication table” or tzolk’in table. They are signed with the two prefixes C and D that, epigrapher David Kelley argued in 1976, are the multiplication signs: C is read as “carries over to” and D as “divides into” or “is the value at” in the running of a multiplication. The Maya, in other words, did not have place-value arithmetic in the elementary-school sense, but they had algorithmic, table-based arithmetic that achieved the same results.

The C and D prefixes also appear in the Long Count Subperiod tables of the codex, which set out the “C” carry value and “D” dividend for cycles like the 9-day, 819-day, 1,296-day, and 46-month counts. The system could be thought of as pre-algorithm in the modern sense: a fixed sequence of operations, with the C and D prefixes acting as instructions, that produced the next number in a table by reading up the previous one. Maya arithmetic was, in modern terms, tabular and not analytic. Scribes did not “solve” equations; they looked up the next entry in a memorized or written table and adjusted it according to a known correction.

The 365-day year and the rounding problem

The haab’ solar year of the Maya is exactly 365 days, while the true tropical year is approximately 365.2422. The difference, about 0.2422 days per year, accumulates to one full day in roughly four years. The haab’ thus drifts about one day in every four years relative to the seasons and the heliacal rising of stars. The drift over 52 haab’ (a Calendar Round) is about 13 days, and the calendar is no longer synchronized with the agricultural seasons within a single lifetime.

The Maya knew this and built corrections into their tables. The Dresden Codex includes correction cycles of 4 × 365 + 1 = 1,461 days (one quarter of a Julian year of 1,461 days? No: 4 × 365 + 1 = 1,461 days, which is the Venus correction for the 5 × 584 = 2,920 day cycle, adjusted for the actual mean synodic period of 583.92 days), and corrections of 1,040 haab’ (379,600 days) and 1,296 haab’ (473,040 days) that re-aligned the rounded 365-day year to the actual tropical year with a residual drift of about 0.001 day per cycle. The same corrections appear in the calendar system pillar and in the page on astronomical cycles and observations.

Lunar series and the Supplementary Series

Maya inscriptions frequently carry a Lunar Series — a block of glyphs that records the 29- or 30-day lunation, the moon’s age in days, the moon number within a six-month, semi-period, and the position of the Moon in the 6-month eclipse warning cycle. The full Lunar Series includes glyphs for “new moon” and “full moon,” for the moon’s age (0 to 29), and for an “8-CR” or “9-CR” coefficient (the position of the lunation in the 9-CR / 8-CR 46-month cycle). The Supplementary Series in which the Lunar Series sits also records the 260-day position (the tzolk’in day) of the given date, the year-bearer (the haab’ day that fell on the first Pop of the year), and the 9-day cycle of the Lord of the Night. The arithmetic embedded in the Lunar Series was thus a tightly integrated set of tables that combined calendars and lunar observations. The Dresden Codex lunar tables are essentially the inscribed version of the same logic.

Limits of the system: what the Maya did not do

The Maya number system, for all its sophistication, was not the equal of the Indian or modern Arabic systems in analytic terms. Maya arithmetic worked by table-lookup, with the C and D prefixes providing the carry/dividend logic; it did not produce a concept of rational number, irrational quantity, or algebraic identity. The Maya did not have a method of iterated division that gave decimal expansions; their calendar corrections were hand-set cycles, not fractions of a day. There is no surviving evidence of a Maya concept of the proof, the derivation, or the theorem. What the system did supremely well was to count days and to predict the moments of astronomically significant events, and at that task — given the observational instruments available — it was remarkably successful.

Sources

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Sharer, R. J. & Traxler, L. P. (2006). The Ancient Maya (6th ed.). Stanford University Press. — link
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